I want to prove
$$ \left| \int_\Gamma F \cdot dl \right| \leq \max_{x \in \Gamma} \left \{ \left| F(x)\right| \right\} \int_\Gamma dl $$
where $F : R^n \rightarrow R^n$ is continuous, $\Gamma \in R^n$ a simple, regular curve and we let $\gamma : [a,b] \rightarrow \Gamma$ be its parametrization:
By definition (the equality),
$$ \left| \int_\Gamma F \cdot dl \right| = \left | \int_a^b F(\gamma(t)) \cdot \gamma'(t) dt \right | \leq \int_a^b \left | F(\gamma(t)) \cdot \gamma'(t) \right | dt $$
Now, I really want to write
$$ \left | F(\gamma(t)) \cdot \gamma'(t) \right | \leq \left | F(\gamma(t)) \right | \left | \gamma'(t) \right | $$
after which the result is immediate, but I'm unable to prove the above. Maybe it's the wrong way to go. (I'm thinking maybe I'll have to use the Cauchy-Schwarz inequality? Can't seem to figure it out..)
Any pointers are appreciated!